On vanishing viscosity approximation of conservation laws with discontinuous flux.

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On vanishing viscosity approximation of conservation laws with discontinuous flux.

Boris Andreianov, Kenneth Hvistendahl Karlsen, Nils Henrik Risebro

To cite this version:

Boris Andreianov, Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On vanishing viscosity approx- imation of conservation laws with discontinuous flux.. Networks and Heterogeneous Media, AIMS- American Institute of Mathematical Sciences, 2010, 5 (3), pp.617-633. �10.3934/nhm.2010.5.617�.

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AIMS’ Journals

VolumeX, Number0X, XX200X pp.X–XX

ON VANISHING VISCOSITY APPROXIMATION OF CONSERVATION LAWS WITH DISCONTINUOUS FLUX.

Boris Andreianov

Universit´e de Franche-Comt´e, 16 route de Gray 25030 Besan¸con Cedex, France

Kenneth H. Karlsen and Nils H. Risebro

Centre of Mathematics for Applications, University of Oslo P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Abstract. We characterize the vanishing viscosity limit for multi-dimensional conservation laws of the form

ut+ divf(x, u) = 0, u|_t=0=u0

in the domainR⁺×R^N. The flux f = f(x, u) is assumed locally Lipschitz continuous in the unknownu and piecewise constant in the space variablex;

the discontinuities off(·, u) are contained in the union of a locally finite number of sufficiently smooth hypersurfaces ofR^N. We define “G_{V V}-entropy solutions”

(this formulation is a particular case of the one of [3]); the definition readily implies the uniqueness and theL¹ contraction principle for theG_{V V}-entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation

u^ε_t+ div (f(x, u^ε)) =ε∆u^ε, u^ε|_t=0=u0, ε↓0,

of the conservation law. We show that, providedu^ε enjoys anε-uniformL^∞ bound and the fluxf(x,·) is non-degenerately nonlinear, vanishing viscosity approximationsu^εconverge asε↓0 to the uniqueG_{V V}-entropy solution of the conservation law with discontinuous flux.

Introduction. The study of conservation laws and related degenerate parabolic problems with space-time discontinuous flux has been intense during the last fifteen years. It is stimulated by applications such as sedimentation, porous medium flows in discontinuous media, road traffic models. We refer to [1]–[11], [13, 14], [16]–

[18], [23]–[26] and references therein for some of the applications and known results.

Notice that only very few studies treat the multidimensional case.

However, most of the interesting phenomena appear already in the model one- dimensional case, with the discontinuity along Σ ={x= 0}:

ut+ (f(x, u))x= 0, f: (x, z)∈R×R7→

(f^l(z) x <0,

f^r(z) x >0. (1) From the purely mathematical viewpoint, the problem is quite challenging because of the possibility to give various non-equivalent generalizations of Kruzhkov’s notion of entropy solution; moreover, different entropy solutions to the same equation may

Date:December 1, 2009.

2000Mathematics Subject Classification. Primary: 35L65.

Key words and phrases. multidimensional hyperbolic scalar conservation law, entropy solution, discontinuous flux, boundary trace, vanishing viscosity approximation, admissibility of solutions.

1

correspond to different applicative contexts. This phenomenon was discovered by Adimurthi, Mishra and Veerappa Gowda in [1]. In [7], B¨urger, Karlsen and Towers proved well-posedness for (1) for a whole class of different solution notions.

Following [7] and the previous works [5, 6], in [3] we set up a framework that encompasses all the notions of solution to the Cauchy problem for (1) which lead to anL¹-contraction semigroup. An interesting application can be found in [2]. The goal of the present note is to provide a separate description of the important par- ticular case of the standard vanishing viscosity limits for (1) and for corresponding multidimensional problems.

Let us give a brief account on the previous work on the subject. Vanishing viscosity limits for conservation laws with discontinuous flux were studied in many preceding works, including those of Gimse and Risebro [13, 14], of Karlsen, Risebro and Towers [16, 17, 18, 25, 26], of Diehl [8, 9, 10, 11], of Panov [23], and many others. In all these works, some intrinsic “entropy” formulations for (1) were given, for which existence and/or uniqueness of solutions was analyzed. The work [23]

contains the most general existence result; see also [17]. Notice that although the definition of solution in [17] is inspired by the vanishing viscosity method, existence can rely upon a justification of convergence of suitably designed numerical schemes (cf. [7] and [3]). The uniqueness issue is most challenging. In [16, 18, 25, 26], the authors give an integral formulation of the Kruzhkov type, with a penalization term supported at the discontinuity hypersurfaces Σ of the flux mapping (t, x)7→f(x,·).

Then uniqueness is justified under the so-called crossing condition; uniqueness may fail when the crossing condition fails (see [3]). Also for the formulation of [23], in general one cannot hope for uniqueness. Diehl, in the works [8, 9, 10] (see also Gimse and Risebro [13, 14]), obtained an entropy formulation on the interface Σ in terms of restrictions on the one-sided limits on Σ of a weak solution u. This

“coupling approach” turns out to be very general, thanks to the strong trace results for entropy solutions (see [22]). The Γ-condition of Diehl [8, 9, 10] was derived from the vanishing viscosity (plus smoothing) standing-waves approach of [12], and expressed in a rather complicated manner. Recently in [11], Diehl reformulated the Γ-condition under a simple form reminiscent of the Ole¨ınik entropy conditions;

and he succeeded in proving uniqueness of solutions for this formulation, without requiring the crossing condition of [18]. Consequently, the Γ-condition of Diehl [11]

should be recognized as the right admissibility condition for the vanishing viscosity limits for (1). Our contribution can be seen as a justification of existence for the Diehl formulation.¹

The description we will give of the “vanishing viscosity germ” GV V (see Defini- tion 1) turns out to be exactly this new form of the Diehl’s Γ-condition. This is by no means surprising. Indeed, our analysis also stems from a simplified vanish- ing viscosity standing-waves analysis (see Proposition 7); then, in order to link the viscosity profiles of Proposition 7(i) to the germG_{V V}, we take advantage of some hints from the general theory of admissibility germs for (1) (see [3] and the Appen- dix of the present paper). As soon as the admissibility germ G_{V V} is identified, we defineG_{V V}-entropy solutions intrinsically. To this end, we either prescribe possible

1In fact, we embed the question of identification of the vanishing viscosity limits into a kind of theory constructed in [3], which also covers different solutions such as those of [1, 7]; with this general point of view, justification of uniqueness is immediate as soon as the properties of the corresponding “admissibility germ” are established. Extension from the model setting (1) to the general multidimensional setting becomes a matter of techniques.

one-sided traces ofuat the discontinuity surface Σ (cf. Diehl [11]); or, we postulate global entropy inequalities not with the Kruzhkov entropiesz7→ |z−k|,k= const, but with “adapted entropies” z 7→ |z−c(x)|, where piecewise constant functions c(x) are defined from the germGV V. The latter approach follows the idea of Baiti and Jenssen [6], of Audusse and Perthame [5] (cf. the interesting re-interpretation of Panov [24]) and of B¨urger, Karlsen and Towers [7].

In our framework, the main restriction on the flux f is the one that ensures a uniformL^∞ bound on solutionsu^ε of equation (13) below. We make a number of simplifying assumptions, including the Lipschitz continuity and the genuine non- linearity of f(x,·) in the sense f⁰(x,·)6= 0 a.e., the smoothness of the discontinuity surfaces off(·, u) and their independence oft. Most of these assumptions can be by- passed; see [3, 4]. For the sake of simplicity, we treat the case of a sole discontinuity off(·, u) along a hypersurface

Σ =(x1, x⁰)∈R^N

x1= Φ(x⁰)

of R^N given by the graph of a smooth function Φ : R^N−1 → R. The case with a locally finite number of smooth discontinuity hypersurfaces (possibly crossing, or piecing together) can be obtained similarly, using partition of unity techniques.

Thus, our result applies, e.g., to conservation laws in stratified media, such as those that appear in geological studies.

Let us give the outline of the paper. In Section 1 we give the definitions (which take the form of two equivalent formulations) and state the main results. In Sec- tion 2, we motivate the definitions in the one-dimensional case (1). Section 3 con- tains the proof of uniqueness and of the equivalence of the two main definitions.

In Section 4, the existence is shown via convergence analysis of the vanishing vis- cosity approximations. An appendix summarizes the framework adopted in [3], and contains one longer proof. We refer to [3, 4] for the details and an extensive bibliography.

1. Vanishing viscosity germ, G_{V V}-entropy solutions and well-posedness.

Let Φ :R^N−1→Rbe aC² function. Denote Ω^l:=R⁺×(x1, x⁰)∈R^N

x1<Φ(x⁰) , Ω^r:=R⁺×(x1, x⁰)∈R^N

x1>Φ(x⁰) ,

and Σ := Ω^l∩Ω^r. Forσ∈Σ, denote byν(σ) the unit vector normal to Σ pointing from Ω^l to Ω^r. We consider fluxes of the form

f: (x, z)∈R^N ×R7→

(f^l(z) x∈Ω^l

f^r(z) x∈Ω^r, f^l,r∈W_loc^1,∞(R), f^lr06= 0 a.e. (2) Forσ∈Σ,f^l,r(σ;·) denotes the normal componentf^l,r(·)·ν(σ) on Σ off^l,r(·).

In order to simplify the presentation, we will make appeal to strong one-sided traces² of a solutionuon Σ.

2Let us stress that the existence of strong tracesof a solutionurelies on the genuine nonlinearity assumption on the fluxesf^l,r; nonetheless, our formulation can be adapted to the case of arbitrary fluxes. In the general case, one works with strong traces of the normal components f^l,r(u)·ν of the flux, and the normal componentsq^l,r(u)·ν of the corresponding Kruzhkov entropy fluxes for a solution u. See Panov [22] for the definition of the relevant trace notion, and [3] for the corresponding formulation which bypasses the existence of the tracesγ^l,ruof the solutionuitself.

We say that a functiong ∈L^∞(R⁺×R^N) admits a right-sided trace γ^rg on Σ in the strong sense (that is, in theL¹_loc topology), if for allξ∈ D(R⁺×R^N),

limh↓0

1 h

Z

R⁺

Z h 0

Z

R^N−1

|g(t, σ+(y1,0))−(γ^rg) (t, σ)|ξ(σ)dtdy1dx⁰ = 0, (3) where σ = (Φ(x⁰), x⁰). The definition of the strong left-sided trace γ^lg on Σ is analogous, withh↓0 replaced byh↑0 in the above formula. The strong traceγ⁰gof gon the set{t= 0}is defined similarly (see e.g., [21]). Note that ifq:R^N×R−→R is continuous andgadmits one-sided tracesγ^l,rgon Σ, thenq◦g:=q(·, g(·)) admits one-sided traces on Σ, and

γ^l,r(q◦g)

(σ) =q(σ,(γ^l,rq)(σ)) H^N a.e. forσ∈Σ.

Now, let us introduce the key object that governs the admissibility of solutions.

Definition 1. For a given couple of functions f^l,r∈C(R), we denote byG_{V V} the set of all couples (u^l, u^r)∈R² satisfying











































s:=f^l u^l=f^r(u^r)and either u^l=u^r,

or ul< ur and there exists a u^o∈

u^l, u^rsuch that







f^l(z)≥sfor allz∈ u^l, u^o, and

f^r(z)≥sfor allz∈[u^o, u^r], or ul> ur and there exists a

u^o∈

u^r, u^lsuch that







f^l(z)≤sfor allz∈ u^o, u^l, and

f^r(z)≤sfor allz∈[u^r, u^o].

(4)

This set is called thevanishing viscosity germ associated with the couple (f^l, f^r).

Remark 2. In [11], Diehl reformulated the Γ-condition of [8, 9, 10] under the following form: A couple (u^l, u^r) satisfies the Γ-condition if

f^l(u^l) =f^r(u^r) and there existsu^o∈ch(u^l, u^r) such that (u^r−u^o) (f^r(z)−f^r(u^r))≥0∀z∈ch (u^r, u^o),

u^o−u^l

f^l(z)−f^l(u^l)

≥0∀z∈ch u^l, u^o ,

(5)

where for a, b∈R, ch(a, b) denotes the convex hull [min{a, b},max{a, b}]. Clearly, (4) coincides with (5). Conditions (4),(5) are reminiscent of the Ole¨ınik admissibility condition (for the case of convex flux functionsf^l,r) and of the “chord condition”

(see e.g., [15] and the pioneering work [12] of Gelfand), since the chord conditions (4) and (5) are derived from the travelling-wave approach [12].

Using the previous notation, we call G_{V V}(σ) the vanishing viscosity germ asso- ciated withf^l,r(σ;·). Now we can defineGV V-entropy solutions.

Definition 3. A functionu∈L^∞(R⁺×R^N) is called aG_{V V}-entropy solution of

ut+ divf(x, u) = 0 (6)

u|t=0=u0 (7)

with fluxfgiven by (2), if

(i) the restriction ofuon Ω^l,r is a Kruzhkov entropy solution of equation (6);

(ii) for H^N-a.e. σon Σ, the couple of strong traces (γ^lu)(σ),(γ^ru)(σ)

of uon Σ belongs to the vanishing viscosity germGV V(σ);

(iii) H^N-a.e. on{0} ×R^N, the initial trace γ⁰uequals u0.

Note that this definition makes sense. Indeed, condition (i) implies the existence of the initial trace γ⁰u (see Panov [21]) and of the boundary traces γ^l,ru on Σ, because Σ is of classC¹ andf^l,r are non-degenerate (see Panov [22]).

Let us give another formulation, which does not involve boundary traces of u.

Forc∈R,

q(x;·, c) := sign(· −c) f(x,·)−f(x, c)

is the entropy flux associated with the Kruzhkov entropy|· −c|. We writeq^l,r(σ;·, c) forq^l,r(·, c)·ν(σ), with the obvious meaning of the superscriptsl, r. We will also use q±(x;·, c) andq^l,r_±(σ;·, c) which correspond to the semi-Kruzhkov entropies (· −c)^±. For (c^l, c^r)∈R², consider

c(x) =c^l1Ω^l(x) +c^r1^Ωr(x). (8) Definition 4. A function u∈ L^∞(R⁺×R^N) is called a GV V-entropy solution of problem (6),(7) with flux f given by (2), if, firstly, it is a solution in the sense of distributions; and secondly, for all couples (c^l, c^r)∈R² and c(x) given by (8), for allξ∈ D(R⁺×R^N),ξ≥0, one has

Z

R⁺

Z

R^N

|u(t, x)−c(x)|ξt+q(x;u(t, x), c(x))· ∇ξ dxdt

− Z

R^N

|u0(x)−c(x)|ξ(0, x)dx+Z

Σ

RV V σ; c^l, c^rξ(σ)dσ≥0, (9) with some “remainder function”RV V : Σ×R²−→R⁺ which is Carath´eodory and fulfills

∀ c^l, c^r

∈ G_{V V}(σ), lim

r↓0− Z

Br(σ)∩Σ

RV V(σ⁰; (c^l, c^r))dσ⁰ = 0, (10) and

∀(c^l, c^r)∈R²and ∀ a^l, a^r

∈ GV V(σ) q^r(σ;a^r, c^r)−q^l σ;a^l, c^l

≤RV V σ; c^l, c^r

. (11)

In [3], the remainder functionRV V is given explicitly; yet the definition does not depend on the choice ofRV V, as soon as the properties (10),(11) are fulfilled.

The equivalence of Definitions 3 and 4 will be shown in Section 3.

Although Definition 4 is not used in the present work, this kind of global entropy formulation would be useful, e.g., for the numerical analysis of the problem (cf. [7, 3, 2]). Indeed, Definition 3 is convenient for the uniqueness proof, but it is not well suited for passage to the limit (cf. the proof of Theorem 5, where the justification of Definition 3(ii) is indirect). On the contrary, it is clear that Definition 4 is stable under theL¹_loc convergence of bounded sequences of solutions.

Under the assumptions on Σ andfstated above, we prove

Theorem 5. (i) Assumeu,uˆare G_{V V}-entropy solutions of (6)with initial data u0,uˆ0∈L^∞(R^N), respectively. Then the following Kato inequality holds: For allξ∈ D(R⁺×R^N),ξ≥0,

Z

R⁺

Z

R^N

(u−u)ˆ ⁺ξt+q+(x;u,u)ˆ · ∇ξ

dxdt+Z

R^N

(u0−uˆ0)⁺ξ(0,·)≥0. (12)

(ii) Let {u^ε}_ε>0 be anL^∞ bounded sequence of solutions to

u^ε_t+ divf(x, u^ε) =ε∆u^ε (13) withu^ε|_t=0=u^ε₀; letu^ε₀→u0inL¹_loc(R^N). Thenu^εconverges a.e. onR⁺×R^N to the uniqueG_{V V}-entropy solution of problem (6),(7)asε↓0.

It is classical that for locally Lipschitz fluxesf^l,r, the Kato inequality (12) gives uniqueness, theL¹ contraction and comparison principles.

It is easy to see that in general, GV V-entropy solutions need not exist. For instance, if for someσ∈Σ, the ranges off^l,r·ν(σ) do not intersect, the Rankine- Hugoniot condition f^l(u^l) = f^r(u^r) cannot hold for any couple (u^l, u^r). In this case, there is no uniformL^∞bound on the sequence of viscous approximationsu^ε. TheL^∞bound can be enforced through different assumptions; e.g., it is enough to have f^l,r(0) = 0_R^N =f^l,r(1) and 0 ≤u0 ≤1. This is the case for the road traffic and for some porous medium models whereuhas the meaning of relative density.

Let us recapitulate our results for this important particular case.

Corollary 6. Let f^l,r be zero at the endpoints of the interval [0,1]. Then for all measurable initial datumu0:R^N 7→[0,1]there exists a uniqueG_{V V}-entropy solution u=:Su0 of problem (6),(7).

The restriction onL¹(R^N; [0,1])of the mapS defined above is an order-preserving semigroup of contractions. Moreover, S is the limit (in the L¹_loc topology) of the solution semigroupsS^ε:u07→u^ε for the vanishing viscosity regularizations (13).

2. Motivations. In this section, we limit our attention to the model one-dimensional problem (1). We first perform a standing-wave analysis of the problem, and then relate the result to the description (4) of the vanishing viscosity germG_{V V}. Proposition 7.

(i) If(u^l, u^r) belongs to the setG_{V V}^o of couples satisfying f^l(u^l) =f^r(u^r) =:s,

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



either u^l=u^r; or ul< ur







f^l(z)> sfor all z∈ u^l, u^r , or

f^r(z)> sfor all z∈ u^l, u^r

, or ul> ur







f^l(z)< sfor all z∈ u^r, u^l

, or

f^r(z)< sfor all z∈ u^r, u^l ,

(14)

then there exists a function W : R → R such that limξ→−∞W(ξ) = u^l, limξ→+∞W(ξ) =u^r, andu^ε(t, x) =W(x/ε)solves (13)inD⁰((0,+∞)×R).

(ii) The sets G=G^o_{V V} andG=GV V fulfill the “L¹D property”

∀(c^l, c^r),(b^l, b^r)∈ G, q^l(c^l, b^l)≥q^r(c^r, b^r). (15) (iii) Assume thatG ⊂R² satisfy (15)and that for all(a^l, a^r)∈ G,f^l(a^l) =f^r(a^r).

Then the inclusion G_{V V}^o ⊂ G implies the inclusionG ⊂ G_{V V}. In particular,

∀ c^l, c^r

∈ G_{V V}^o , f^l a^l=f^r(a^r) & q^l a^l, c^l

≥q^r(a^r, c^r)

⇓ a^l, a^r

∈ G_{V V}.

(16)

Proof (sketched). (i) In the caseu^l=u^r, the standing-wave profileW can be chosen constant onR. The four other cases are symmetric. For instance, in the caseu^l< u^r and f^l(z)> s for all z ∈(u^l, u^r], the profile W is a continuous function constant (equal tou^r) on [0,+∞). On the interval (−∞,0],W is constructed as the maximal solution of the autonomous ODE W⁰ =f^l(W)−f^l(u^l) with the initial condition W(0) = u^r. Indeed, becausef^l(w)−f^l(u^l) =f^l(w)−s >0 for w∈(u^l, u^r], the solutionW is non-decreasing. Becausef^lis assumed Lipschitz continuous andu^lis a stationary solution,W is defined on the whole interval (−∞,0], and there exists d:= limξ→−∞W(ξ)∈[u^l, u^r]. In this case,f^l(d)−s= 0, which yieldsd=u^l. The result is easy to prove also in the case of merely continuous functionsf^l,r (see [3]).

(ii) One can prove this claim by a tedious case study; see [3]. Let us give an argument that uses the structure of the solutions of (13). Notice that (15) for G=G^o_{V V} can also be deduced from the Kato inequality for solutions of (13). More precisely, let (c^l, c^r),(b^l, b^r)∈ G_{V V}^o . According to (14), letu^ε(t, x) :=W(x/ε) with W(−∞) =c^l, W(+∞) =c^r; similarly, let ˆu^ε(t, x) := ˆW(x/ε) with ˆW(−∞) =b^l, Wˆ(+∞) =b^r. Then one shows the Kato inequality:

Z

R⁺

Z

R

(|u^ε−uˆ^ε|ξt+q(x;u^ε,uˆ^ε)ξx+ε|u^ε−uˆ^ε|ξxx)dxdt≥0. (17) for allξ∈ D((0,∞)×R),ξ≥0. Lettingε→0, we have

u^ε(t, x)→c^l1x<0+c^r1x>0 and ˆ

u^ε(t, x)→b^l1x<0+b^r1x>0. Therefore from (17), we readily get

q^l c^l, b^l

−q^r(c^r, b^r) Z

R⁺

ξ(t,0)dt≥0

by the Green-Gauss theorem. Therefore (15) follows forG =G_{V V}^o . Then theL¹D property (15) forG =G_{V V} is inferred; indeed, G_{V V} turns out to be the closure of G_{V V}^o in the sense defined in [3] (see also the appendix in Section 5), and the closure operation preserves theL¹Dproperty (15).

(iii) The proof (taken from [3]) is postponed to the appendix in Section 5.

Remark 8. For givenf^l,rthere may exist many different subsetsGofR²satisfying theL¹D property (15) and the equalities∀(c^l, c^r)∈ G f^l(c^l) =f^r(c^r) (these equal- ities encode the Rankine-Hugoniot condition on Σ). Such G is called a maximal L¹D admissibility germ if it possesses no nontrivial extension satisfying the same properties. Any maximal germ leads to a notion ofG-entropy solution (see [3]).

Proposition 7 (ii) and (iii) mean that the germ GV V is maximal. Proposi- tion 7 (iii) also states that G = G_{V V}^o admits a unique maximal extension. This implies, e.g., that in the constraints (10) and (11) of Definition (3), GV V could be replaced withG_{V V}^o .

In the model case (1), we can simplify Definition 4 by setting, regardless ofσ∈Σ, RV V σ; c^l, c^r:=M dist c^l, c^r

,G_{V V}^o

, (18)

where dist is the Euclidean distance on R² and M is a sufficiently large positive constant.

Now let us explain the notion of aGV V-entropy solution. Both Definitions 3 and 4 state the Kruzhkov entropy inequalities locally, away from the flux discontinuity interface Σ. But they also contain a description of the coupling of u|Ω^l and u|Ω^r

across Σ. The idea behind Definition 3 lies in the identification of the possible trace couples (γ^lu, γ^ru) of admissible solutionsu. In turn, Definition 4 (withRV V given by (18)) explicitly allows for selected “elementary” weak solutions to (1):

c(x) =c^l1{x<0}+c^r1{x>0},

which play the role of the constants in the classical Kruzhkov formulation. The definitions are inspired by the idea of “adapted entropies” (cf. Baiti and Jenssen [6], Audusse and Perthame [5], B¨urger, Karlsen and Towers [7]).

The selection of the elementary solutions that should be admitted is based upon the vanishing viscosity approach. Proposition 7(i) means that c(·) correspond- ing to (c^l, c^r) ∈ G_{V V}^o should be admitted in Definition 4, and the trace couples (c^l, c^r)∈ G^o_{V V} should be admitted in Definition 3. Indeed, in this casec(·) is clearly obtained as the limit of the viscous standing-wave profiles u^ε, moreover, we have γ^l,r(c(·))(t,0) =c^l,r for allt >0.

Proposition 7(ii) implies the dissipativity property for the coupling ofu|Ω^l and u|Ω^r across Σ. This property ensures the Kato inequality (12) and yields the unique- ness ofGV V-entropy solutions.

Reciprocally, property (16) of Proposition 7(iii) constrains the traces (γ^lu, γ^ru) of an arbitrary function u obtained as limit of viscous approximations u^ε, thus giving rise to Definition 3(ii). Indeed, a Kato inequality holds for any pairu^ε,uˆ^εof solutions of (13); this inequality, “inherited” at the limit, yields the Kato inequality (12) for any pair of viscous limits u,u; and the (elementary) solutions ˆˆ u(t, x) = c(x) =c^l1{x<0}+c^r1{x>0}, (c^l, c^r)∈ G^o_{V V}, have already been identified as viscous limits. From (12) and (16) we derive that (γ^lu, γ^ru)(t)∈ GV V, for a.e. t.

3. The uniqueness proof and equivalence of definitions. Throughout this section, we fix a non-negative non-increasing (truncation) function ξ∗ in D(R⁺) satisfying

ξ∗(s) =

(1 s <1,

0 s >2, and we setξh(x) =ξ∗

|x1−Φ(x⁰)|

h

. Proposition 9. Definitions 3 and 4 are equivalent.

Proof. It is standard (see in particular Panov [21]) that Definition 3(i),(iii) is equivalent to inequalities (9) withξ∈ D(R⁺×(R^N\Σ)),ξ≥0.

For a generalξ∈ D(R⁺×R^N), ξ≥0, we have ξ(1−ξh)∈ D(R⁺×(R^N \Σ)).

Thus we can focus on the contribution of the truncated test functionξξh into (9).

We only have to show that Definition 3(ii) is equivalent to the statement that, for all pairs (c^l, c^r)∈R², the inequality

lim inf

h↓0

Z

R⁺

Z

R^N

ξq(x;u, c(x))· ∇ξhdxdt+Z

Σ

RV V(σ; (c^l, c^r))ξ(σ)dσ≥0 (19) holds. The existence of strong tracesγ^l,ru(which follows from [22] and assumption (2)) and the definition ofξhallows us to reformulate (19) as

Z

Σ

q^l γ^lu, c^l

−q^r(γ^ru, c^r) +RV V(σ; (c^l, c^r))ξ(σ)dσ≥0, (20) for all pairs (c^l, c^r).

Now, assume Definition 3(ii) holds. As soon as (11) is guaranteed, (20) fol- lows from (11) and Proposition 7(ii). Therefore it is sufficient to construct a

Carath´eodory functionRV V satisfying (10) and (11). In the case of a flat interface Σ, one can take the expression (18). A more subtle choice is

RV V σ; c^l, c^r

= 2 inf_(bl,b^r)∈GV V(σ)

Osc f^l(σ;·) ;c^l, b^l+ Osc (f^r(σ;·) ;c^r, b^r) , where Osc(g;c, b) denotes the oscillation of the function g on the segment with endpointsb,c. We refer to [3] for the details concerning the choice ofRV V(·; (c^l, c^r)).

Reciprocally, assume (20) with RV V satisfying (10) and (11). Lettingξ|Σ con- centrate at a Lebesgue pointσofγ^l,ru, with the help of (10) we find that

for all (c^l, c^r)∈ G_{V V}(σ) q^l((γ^lu)(σ), c^l)−q^r((γ^ru)(σ), c^r)≥0.

By (16) we conclude that ((γ^lu)(σ),(γ^ru)(σ))∈ G_{V V}(σ).

Proof of Theorem 5(i). We use Definition 3. From (i) and (iii), by the standard Kruzhkov doubling of variables technique we obtain the Kato inequality (12) with ξ∈D(R⁺×(R^N \Σ)). As in the previous proof, using the truncationξh, we see that it is sufficient to prove that

lim inf

h↓0

Z

R⁺

Z

R^N

ξq+(x;u,u)ˆ · ∇ξhdxdt≥0. (21) The definition ofξh and the existence of the strong tracesγ^l,ruand γ^l,ruˆ allow to rewrite (21) asq+^l(γ^lu, γ^lu)ˆ ≥q+^r(γ^ru, γ^ru),ˆ H^N-a.e. on Σ. This inequality is easily checked from Definition 3(ii) and theL¹D property (15) ofG_{V V}(σ),σ∈Σ.

4. Convergence of the vanishing viscosity method. In the model case (1), the outline of the proof is given at the end of Section 2. In the general case, we also exploit the Kato inequality for solutionsu^ε and ˆu^ε, but we have to deal with solutions to the nonhomogeneous equation (13). A blow-up technique yields the conclusion.

Proof of Theorem 5(ii). First, the L^∞ bound assumed on u^ε and the genuine nonlinearity assumption in (2) allow us to use the precompactness results of Lions, Perthame and Tadmor [19] or of Panov [20, 23] in the domains Ω^l,r. Hence, up to extraction of a convergent sequence,u^εconverges a.e. to someu∈L^∞(R⁺×R^N).

Moreover,ufulfills Definition 3(i), (iii); it is also a solution of (6) in the sense of distributions, so that the Rankine-Hugoniot condition on Σ holds. As soon as we prove thatusatisfies Definition 3(ii), by the uniqueness result of Theorem 5(i) we get the convergenceu^ε→uasε↓0

As mentioned in the introduction, Definition 3(i) and the flux non-degeneracy in (2) ensure the existence of the strong traces γ^l,ruon Σ. Letσo = (to, xo) be a common Lebesgue point ofγ^l,ru. The Rankine-Hugoniot condition foruimplies

f^l γ^lu (σo)

·ν(σo) =f^r_o((γ^ru) (σo))·ν(σo). In order to conclude the proof, we only have to justify that

q^l γ^lu

(σo), c^l

·ν(σo)≥q^r((γ^ru) (σo), c^r)·ν(σo), (22) for all pairs (c^l, c^r)∈ G_{V V}^o (σo). Indeed, (22) and property (16) would yield

γ^lu(σo), γ^lu(σo)

∈ GV V(σo).

Recall thatf^l,r(σ;·) denotesf^l,r(·)·ν(σ); we will also writef_o^l,r(·) forf^l,r(σo;·).

Translating and rotating the axes, we can (at least, locally) reduce the situation to

xo= 0, Φ(0) = 0 and ∇Φ(0) = 0, (23)

so that {(t, x1, x⁰)|x1 = 0} is the tangent plane to Σ at the point σo= (t,0). By Proposition 7, there exists a solution to the one-dimensional problem

(fo(x1, W))_x1 =Wx₁x₁, W(−∞) =c^l, W(+∞) =c^r (24) (W is the standing-wave profile corresponding to the model problem (1) with f_o(x1,·) =f_o^l(·)1{x1<0}+f_o^r(·)1{x1>0}). The properties ofW include

W ∈C(R)∩W^2,∞(R\{0}),

W⁰ ∈L¹(R)∩L^∞(R), W⁰⁰|_R\{0}∈L¹(R\{0}), W⁰(0⁺)−W⁰(0⁻) =f_o^r(W(0))−f_o^l(W(0)).

(25)

Consider the approximate solutionsw^ε, ε >0, to equation (13) and their limitw:

w^ε(t, x) := Wx1−Φ(x⁰) ε

, w(t, x) := lim

ε↓0w^ε(t, x) =c^l1Ω^l+c^r1^Ωr.

(26)

Straightforward calculation using the pointwise formulation of (24) and the jump condition in (25) shows that the functionw^ε verifies the equation

w_t^ε+ divf(x, w^ε) =r^ε+ε∆w^ε (27) (in the sense of distributions) with source termr^ε=r1^ε+r2^ε+r^ε3+r^ε4+r^ε5, where

r^ε1=−1

εW⁰(ξ)f⁰(x, W(ξ))· ∇_(x1,x⁰)Φ(x⁰), r2^ε=−1

ε|∇Φ(x⁰)|² W⁰⁰(ξ), r^ε3= ∆Φ(x⁰)W⁰(ξ),

and the termsr₄^ε,r^ε₅ are measures supported on Σ and acting onϕ∈C(R^N+1) by hr^ε4, ϕi:=− W⁰ 0⁺

−W⁰ 0⁻ Z

R⁺

Z

R^N−1

|∇Φ(x⁰)|² ϕ(t,Φ(x⁰), x⁰)dtdx⁰, hr^ε5, ϕi:=Z

R⁺

Z

R^N−1

f^r(W(0))−f^l(W(0))

· ∇_(x1,x⁰)Φ(x⁰)ϕ(t,Φ(x⁰), x⁰)dtdx⁰. In the expressions r₁^ε, . . . , r^ε₃,ξ=ξ(x1, x⁰) = (x1−Φ(x⁰))/ε. The functionsW, W⁰ andW⁰⁰are evaluated pointwise, forξ6= 0;∇_(x1,x⁰₎Φ(x⁰) denotes theN-dimensional vector (0,∇Φ(x⁰)); andf⁰ is the a.e. defined derivative inzoff(x, z). Note that the productW⁰(ξ)f⁰(x, W(ξ)) makes sense.

Taking a smooth approximationHα(u^ε−w^ε) of sign(u^ε−w^ε) for the test function in the difference of equations (13) and (27), as α ↓ 0 we deduce the following Kato inequality: For all non-negative test functions ϕ∈ D(R^N+1) supported in a neighbourhood ofσo,

− Z

R^N+1

(|u^ε−w^ε|ϕt+q(x;u^ε, w^ε)· ∇ϕ+ε|u^ε−w^ε|∆ϕ)dxdt

≤ Z

R^N+1

(|r₁^ε|+|r₂^ε|+M|∆Φ(x⁰)|)ϕ dx1dx⁰dt +M

Z

(t,x⁰)∈R^N

|∇Φ(x⁰)|²+|∇Φ(x⁰)|

ϕ(t,Φ(x⁰), x⁰)dtdx⁰, (28)

where

M = max

W⁰(0⁺)−W⁰(0⁻) ,

f^r(W(0))−f^l(W(0))

,kW⁰k_∞ .

In the sequel, M denotes a generic constant depending on the profile W and on sup|f⁰|on the segment with endpointsc^l,r. Because we havedx1dx⁰=εdξdx⁰ in the sense of measures, the integrability properties in (25) yield∀ϕ∈ D(R^N+1),ϕ≥0,

Z

R^N+1

(|r₁^ε|+|r₂^ε|)ϕ(t, x1, x⁰)dx1dx⁰dt

≤ Z

R^N+1

|W⁰(ξ)| |∇Φ(x⁰)|+|W⁰⁰(ξ)| |∇Φ(x⁰)|²

ϕ(t, ξ, x⁰)dtdξdx⁰

≤M Z

(t,x⁰)∈R^N

|∇Φ(x⁰)|+|∇Φ(x⁰)|²

maxϕ(t,·, x⁰)dtdx⁰.

(29)

Now we fix a test function of the formϕ(t, x) :=ψ(t, x⁰)ξh(x), where ξh was intro- duced in Section 3. Keepinghandψfixed, we letε↓0 in (28). Using the uniform inεbound (29) and the definitions ofuandw, we infer that

− Z

R^N+1

(|u−w|(ψξh)_t+q(x;u, w)· ∇(ψξh))dxdt

≤M Z

R^N

|∇Φ|+|∇Φ|²+h|∆Φ|

ψ dtdx⁰. Now replace ψ by a nonnegative test function ψδ ∈ D(R^N) with integral equal to one, supported in aδ-neighbourhood of σo (here, we mean that Σ is parametrized by (t, x⁰)∈R^N). Ash↓0 and thenδ↓0, the right-hand side of the above inequality vanishes, due to the normalization (23). As to the left-hand side, it converges to

−lim

δ↓0

Z

Σ

q^l γ^lu(σ), c^l

−q^r((γ^ru) (σ), c^r)

·ν(σ)ψδ(σ)dtdx⁰

=− q^l γ^lu(σo), c^l

−q^r((γ^ru) (σo), c^r)

·ν(σo). This establishes (22) and concludes the proof.

Proof of Corollary 6 (sketched). Existence for (13) with u0 ∈ L²(R^N) can be obtained by the classical Galerkin method. Uniqueness and, more generally, the comparison principle and the L¹ contraction property for solutionsu^ε of (13) are also classical (cf. (28) in the above proof). Then the comparison principle allows to drop the restriction onu0for the existence of a solutionu^εto (13).

Finally, because we assume that f^l,r(0) = 0 = f^l,r(1) and 0 ≤ u0 ≤ 1, the comparison principle yields 0≤u^ε≤1. This justifies Corollary 6.

5. Appendix: theory of germs and the maximality of GV V. Here we justify Proposition 7(iii). To this end, let us first give a general definition of anL¹D germ and of the closure operation on germs. In relation with the left- and right-side fluxes f^l andf^r in (1) and the associated Kruzhkov fluxes

q^l,r(z, k) = sign (z−k) f^l,r(z)−f^l,r(k) , we introduce the following definitions:

Definition 10. Aright (respectively, left)contact shock is a couple of real values (u^r, u+) (resp., (u−, u^l)) such that the function

u(x) =u^r1x<0+u+1x>0 (resp.,u(x) =u−1x<0+u^l1x>0)

is a stationary Kruzhkov-admissible shock for the conservation lawut+ (g(u))x= 0 with the fluxg=f^r(resp.,g=f^l).

Definition 11 (Germs; closed, complete, maximal and definite germs).

• Any setGof couples (c^l, c^r)∈R×Rsatisfying the Rankine-Hugoniot relation

f^l(c^l) =f^r(c^r) (30)

and the L¹-dissipativity relation (15) is called anL¹D admissibility germ (a germ, for short) associated with the couple of fluxes (f^l, f^r).

• The closure of a germ G is the smallest set G containing G such that G is topologically closed, and moreover, for all couples (c^l, c^r)∈ G,Galso contains all couples (c−, c+) such that (c−, c^l) is a left contact shock, (c^r, c⁺) is a right contact shock.

• A germGis called closed, ifG=G.

• A germ G is called complete³, if all Riemann problem for (1) admits a self- similar solution usuch that (γ^lu, γ^ru)∈ G, where γ^lu, resp.γ^ru, is the limit ofuasx→0⁻, resp.as x→0⁺.

• We say thatG⁰ is anextensionof a germG ifG ⊂ G⁰ andG⁰ still satisfies the L¹-dissipation property (15) and the Rankine-Hugoniot condition (30).

• A germGis called maximal, if it does not admit a nontrivial extension.

• A germGis called definite, it it admits only one maximal extension.

In relation with definite and maximal germs, consider one more definition.

Definition 12 (dual of a germ). LetG be a germ. The dual ofG is the set G^∗:=

( b^l, b^r

∈R×R

f^l b^l=f^r(b^r) and

∀ c^l, c^r

∈ G, q^l c^l, b^l

≥q^r(c^r, b^r) )

. (31)

We pause to give an example illustrating these definitions. Let f^l(u) = 3u(1−u) andf^r(u) = 4u(1−u).

Foru^r andu^lin [0,1] the right and left contact shocks are given by u+=

(1−u^r oru^r foru^r∈[0,1/2], u^r foru^r∈[1/2,1].

u−=

(u^l foru^l∈[0,1/2], 1−u^l oru^l foru^l∈[1/2,1].

The Rankine-Hugoniot condition implies that any couple (c^l, c^r) in a germ must satisfy

c^r= 1 2

1±q

1−3c^l(1−c^l)

=:h^±(c^l). (32)

In addition, for every two couples (b^l, b^r), (c^l, c^r) in a germ, theL¹Dcondition (15) implies that

(eitherf^l,r b^l,r=f^l,r c^l,r orb^l< c^l =⇒ b^r< c^r.

3The definition in [3] of a complete germ is slightly different; it autorizes left- and right- contact shocks in the solutions of a Riemann problem. Contrarily to [3], the definition of the present paper implies that a complete germ is closed; this is not always convenient.

In particular, in a germ you cannot “jump decreasingly (in the sense thatb^l> b^r) through 1/2 more than once”. Furthermore, this decreasing jump must occur at the maximal allowed value of the flux at the jump.

Hence, an example of a germ is the set G=

b^l∈[0,1/4], b^r=h⁻ b^l) ∪

b^l∈[5/6,1], b^r=h⁺ b^l . By adding all contact shocks toG we obtain its closure,

G=

b^l∈[0,1/4], b^r=h⁻ b^l) ∪

b^l∈[5/6,1], b^r=h⁺ b^l

∪

b^l∈[0,1/4], b^r=h⁺ b^l .

Consider the Riemann problem with left state 3/8 and right state h⁻(3/8). This couple is not in G, and if we wish to find a self similar solution with traces inG, we must first jump by a shock with negative speed to a value c^l ∈[5/6,1]. If the solution is to have traces in G then the trace from the right must be h⁺(c^l). It is however impossible to connect h⁺(c^l) with h⁻(3/8) by a Kruzhkov-admissible solution having waves of non-negative speeds. ThusG is not complete.

Forκ∈[1/4,1/2] we can define a family of maximal extensions toG by G_κ=

b^l, h⁻ b^l

0≤bl≤κ ∪

b^l, h⁺ b^l

0≤bl≤κ

∪

1−κ, h⁻(1−κ) ∪

b^l, h⁺ b^l

1−κ≤b^l≤1 .

Each of these extensions “jumps decreasingly through 1/2” once, and limits the maximal flux through x= 0 to f^l(κ). Since G has several maximal extensions, it is not definite, see Proposition 14(iii) below. Regarding the dualG^∗, by Proposi- tion 13(v), it will not be a germ. For each κ, G_κ = G_κ and we have added pre- cisely the decreasing jump which makes it complete. Hence, by Proposition 15(ii), G_κ^∗=G_κ.

The dual ofG (and also the dual of G) is formed by the addition of all points which satisfy the Rankine-Hugoniot condition and theL¹D condition with respect to points inG(and in G). Hence the dual is given by

G^∗=G^∗=

b^l∈[0,3/4], b^r=h⁻ b^l) ∪

b^l∈[0,1], b^r=h⁺ b^l . Observe that in accordance with Proposition 13(ii)G^∗=∪κ∈[1/4,1/2]Gκ. These sets are depicted in Figure 1.

We refer to [3] for details, further examples and for the proofs of the below relations between different properties ofG,G,G^∗. These propositions can be helpful in order to determine whether a given subset G of R² is a germ, and in order to describe the properties of a given germG.

Proposition 13(dual germ, maximality and definiteness). Let Gbe a subset of R²; letG^∗ be defined by (31).

(i) One hasG ⊂ G^∗ if and only ifG is a germ.

(ii) AssumeGis a germ. ThenG^∗is the union of all extensions ofG. In particular, if G is a definite germ, thenG^∗ is the unique maximal extension ofG.

(iii) One hasG^∗=G if and only ifG is a maximal germ.

(iv) If G is a definite germ, then(G^∗)^∗=G^∗. (v) If G^∗ is a germ, thenG is definite.

Proposition 14 (closure and closed germs).

(i) AssumeGis a germ. Then its closureGis also a germ (thus,Gis an extension of G); furthermore,G ⊂ G^∗ and(G)^∗=G^∗.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b^l b^r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b^l b^r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b^l b^r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b^l b^r

Figure 1. The Rankine-Hugoniot condition shown as broken curves, the other sets as solid curves in the (b^l, b^r) plane. Top left:

the germ G, top right: the closure G, bottom left: the extension G_3/8, bottom right: the dualG^∗.

(ii) If G is a maximal germ, then it is closed.

(iii) Any maximal extension ofG containsG. In particular,G is a definite germ if and only ifG is a definite germ.

(iv) LetG be a definite germ. ThenG-entropy andG-entropy solutions coincide.

Proposition 15 (complete germs).

(i) AssumeG is a complete germ. ThenG is a maximal (and thus closed) germ.

(ii) AssumeG is a germ such thatG is complete. ThenGis definite, andG^∗=G.

Remark 16. Notice that in case (ii) of Proposition 15, G is a definite germ, and G^∗ is maximal and complete. Such germs are expected to lead to a well-posedness theory for G-entropy solutions. The germ GV V of Definition 1 is one example of a maximal germ; it is complete, e.g., under the assumptions of Corollary 6.

In terms of the above definitions, the statement of Proposition 7(iii) exactly means that G_{V V}^o is a definite germ of which G_{V V} is the dual; in particular, G_{V V} is a maximal germ. The below proof is based on the property thatG_{V V} coincides with the closureG_{V V}^o ofG^o_{V V}. Let us point out that the difference between a germ and its closure is responsible for the apparent distinction between the pioneering

“minimal jump” admissibility condition of Gimse and Risebro [13, 14] and the Γ- condition given by Diehl in [8, 9, 10, 11]. The set of trace values determined by the two conditions has the same closure; according to Proposition 15, this distinction does not change the germ-based notion of entropy solution.